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تسجيل مستخدم جديدGlobal fluctuations for 1D log-gas dynamics. (2) Covariance kernel and support
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Jeremie Unterberger
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We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, $ dlambda_t^i=frac{1}{sqrt{N}} dW_t^i - V(lambda_t^i) dt+ frac{beta}{2N} sum_{j ot=i} frac{dt}{lambda^i_t-lambda^j_t}, qquad i=1,ldots,N, $ with $beta>1$, sometimes called generalized Dysons Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $beta$-ensemble, with sufficiently regular convex potential $V$. The limit $Ntoinfty$ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown by the author to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation. We prove a series of results concerning either the Mc Kean-Vlasov equation for the density $rho_t$, notably regularity results and time-evolution of the support, or the associated hydrodynamic fluctuation process, whose space-time covariance kernel we compute explicitly.
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We study in this article the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, begin{equation} dlambda_t^i=frac{1}{sqrt{N}} dW_t^i - V(lambda_t^i) dt+ frac{beta}{2N} sum_{j ot=i} frac{dt}{lambda^
i_t-lambda^j_t}, qquad i=1,ldots,N, end{equation} with $beta>1$, sometimes called generalized Dysons Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $beta$-ensemble, with sufficiently regular convex potential $V$. The limit $Ntoinfty$ is known to satisfy a mean-field Mac-Kean-Vlasov equation. We prove that, for suitable initial conditions, fluctuations around the limit are Gaussian and satisfy an explicit PDE. The proof is very much indebted to the harmonic potential case treated in Israelsson cite{Isr}. Our key argument consists in showing that the time-evolution generator may be written in the form of a transport operator on the upper half-plane, plus a bounded non-local operator interpreted in terms of a signed jump process. As an essential technical argument ensuring the convergence of the above scheme, we give an $N$-independent large-deviation type estimate for the probability that $sup_{tin[0,T]}max_{i=1,ldots,N} |lambda_t^i|$ is large, based on a multi-scale argument and entropic bounds.
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